Lepton Flavor Violating Photoleptonic Effect

a r X i v :0705.2900v 1 [h e p -p h ] 21 M a y 2007Lepton Flavor Violating Photoleptonic E?ect

Edson Carquin,?Yuri Ivanov ?,?Sergey Kovalenko,§and Ivan Schmidt ?

Centro de Estudios Subat′o micos(CES),

Universidad T′e cnica Federico Santa Mar′?a,

Casilla 110-V,Valpara′?so,Chile

We study lepton ?avor violating analogs of the photoelectric e?ect,with a ?nal μor τinstead of

an electron:γe →μand γe →τ.On the basis of the general parametrization of the matrix element

of the electromagnetic current we estimate the upper limits for the cross sections and event rates of

these processes,imposed by the current experimental bounds on μ→eγand τ→eγdecays.

PACS numbers:PACS numbers:Keywords:I.INTRODUCTION Lepton Flavor Violation (LFV)has become an experimental fact after the observation of neutrino oscillations,and this leads immediately to the conclusion of the existence of LFV processes in the sector of charged leptons.However,the amount of LFV transmitted from the neutrino sector to the sector of charged leptons by Standard Model (SM)loops is extremely small,leaving no chance for their experimental observation.On the other hand interactions beyond the SM can potentially induce LFV directly in the sector of charged leptons.Therefore,any observation of an LFV transition of charged leptons would be a signal of physics beyond the SM.This is the motivation for the theoretical and experimental studies of LFV processes with charged leptons.Among them the most attention has been paid to muon-electron nuclear conversion,muon(electron)-nucleon scattering as well as to LFV in decays of mesons,muon,tau (for reviews see,for instance,Refs.[1]).It was recognized that di?erent processes may have quite di?erent sensitivity to the LFV and can shed light on complementary aspects of the underlying physics being,in general,dependent on di?erent combinations of fundamental parameters of new physics.Therefore,looking for new processes potentially capable to render new information on the origen of LFV represents an important quest both for theory and experiment.In the present work we examine a new class of the LFV processes in the charged lepton sector,induced by real photon beams.This is the LFV version of the photoelectric e?ect,with a muon or tau in the ?nal state instead of the usual photoelectron.Our analysis is based on the general parametrization of the electromagnetic current in terms of LFV form factors,without any reference to the underlying physics behind the LFV electromagnetic transitions of leptons.The same form factors describe μ→eγand τ→eγdecays and,therefore,they are limited by the existing results on the experimental searches for these processes.We use these limits to predict upper bounds on the total cross section and event rate of γe →μand γe →τfor the initial electrons bound to atoms.II.MATRIX ELEMENT OF LFV PHOTO-LEPTONIC EFFECT

The amplitud for the transition γl i →l f ,induced by real photons,can be written in the following standard form

?M fi = e μ l f |J μem

(x )|l i e ?ik ·x d 4x,(1)where e λis the photon polarization 4-vector.

2

The most general form of the leptonic matrix element in Eq.(1),consistent with Lorentz covariance and

conservation

of electric current,is

l f |J μem (x )|l i =ˉψf (x ) (f fi E 0(?k 2)+γ5f fi M 0(?k 2))γν g μν??k ν?k μm e

ψi (x )(2)≡ˉψf (x )Γμfi (?k )ψi (x ),

where ψi (f )are the wave functions of the initial(?nal)lepton.We de?ned ?k μ=i (←??μ+?→?μ)the di?erential operator

of 4-momentum transfer,with the ?rst derivative acting to the left hand side and the second one to the right hand side.For the case of free leptons this operator is to be replaced as ?k

μ→k μ=p (f )μ?p (i )μ,where p (i )μand p (f )μare 4-momenta of the initial and ?nal leptons respectively.For convenience we also introduced the function Γμ

fi .In

Eq.(2)the functions f E 0(k 2),f M 0(k 2)and f E 1(k 2),f M 1(k 2)are the conventional monopole and dipole electric and magnetic transition form factors.From T-invariance it follows that all the above form factors are real and symmetric

f if E 0=f fi E 0,f if M 0=f fi M 0,f if E 1=f fi E 1,f if M 1=f fi M 1.(3)

Thus,the same set of form factors describe γl i →l f and l i →l f γprocesses.The monopole form factors must satisfy the ?niteness conditions

f if E 0(0)=f if M 0(0)=0(4)

and,therefore,they do not contribute to the γl i →l f processes with a real photon,which has k 2=0.

Substituting the expression (2)into Eq.(1)and integrating by parts we obtain

?M fi = ˉψf (x )e μΓμfi (k )ψi (x ),e ?ik ·x d 4x (5)

where the vertex function Γμis the function de?ned in Eq.(2).

Let us turn to the LFV photoe?ect:γe →l with l =μ,τ.In this case the initial lepton is the electron bound to the atom with energy εe =m e ?I ,where I is the corresponding value of the ionization energy.The incident real photon with energy ωand momentum k hits the atomic electron and creates the ?nal lepton l with energy εl and momentum p l .Therefore,we can rewrite Eq.(1)in the 3-dimensional transversal gauge,e μk μ=0with e μ=(0,e ),in the form

?M

le =2πδ(εi +ω?εl ) ˉψl (x )(e ·Γ)l ψe (x )e i kx d 3x ≡2πδ(εi +ω?εl )M le ,(6)where ψe,l (x )are the spacial wave functions of the initial electron and ?nal lepton.Here we also introduced the reduced matrix element M le of γe →l transition.In virtue of Eq.(4)the product of vertex function and the photon polarization vector is given by

(e ·Γ)l =i

√

2m e γ0γ? u e 2m e ψ0,(9)

where u e is bispinor amplitude of the electron in the rest frame,normalized by ˉu e u e =2m e .

3 We write the wave function of the?nal lepton in the form

ψl=1

2εl u l e i p l·r+ψ(1) ,(10)

where the termψ(1)represents the leading Ze2Coulomb correction.Its Fourier transform is[2]

ˉψ(1)

?k

= d3xˉψ(1)(x)e i kx=4πZe2ˉu l2εlγ0+γ·(k?p l)

(εl m e)1/2(k?p l)2

ˉu l A l u e(12) with

A l=a(Γ·e)l+(Γ·e)lγ0(γ·b)+(γ·c)γ0(Γ·e)l(13) where

a=1

m e

1

2m e(p l?k)2,c=

k?p l

ω(k?p l)4|p l|Tr (γ0εl?γp l+m l)A l(γ0+1)γ0A?lγ0 d?.(15)

Carrying out the trace one can obtain the following expression

Tr[...]=

8

t2?1in the following form

σ(γe→l)=16α6em Z5m5e

u

F(t,u,v),(19)

where

F(t,u,v)=P(t,u,v)

2uv(u2?v2)2log

u?v

(u2?v2)2

64u4+80u3?10u2?32u?32uv log u?v

III.EXPERIMENTAL CONSTRAINTS ON THE FORM F ACTORS

Since the same form factors f if E1,M1determine bothγl i→l f and l f→l iγprocesses,we can derive upper limits on and f eτE1,M1from the existing experimental bounds onμ→eγandτ→eγ[3,4,5]

f eμ

E1,M1

Br(τ?→e?γ)=Γ(τ?→e?γ)

Γμ≤1.2×10?11(24) and then apply these limits for the evaluation of upper bounds on the processes in which we are interested:γe→μandγe→τ.In Eqs.(23),(24)we useΓτ=2.26×10?5MeV andΓμ=3×10?16MeV for total decay widths of the τandμ.

The decay rates are given by

Γ(l?→e?γ)=m3l

A Fγ×4.35×10?16fb?1/s,(28) where A,Z andκare target material nuclear mass number in atomic units,atomic number and the conversion length in g·cm?2,respectively.The photon?ux Fγis measured in s?1.

As an example we consider a lead(Pb)target with A=207.2,Z=82andκ=7.46g·cm?2.Its corresponding luminosity is

L P b=1.3×10?15·Fγfb?1/s,(29) With this luminosity we estimate the number of the LFV events

R(γe→μ)≈2.0×10?40·Nγforω=1GeV(30)

R(γe→τ)≈1.0×10?40·Nγforω=5GeV(31)

(32) where Nγis the number of the photons absorbed in the lead target.This result means that the observation of one LFV event would require a photon energy deposit to the target of about1030J.It is clear that these conditions are unrealistic.Higher event rates correspond to photon energies too high to be achieved in near future experiments with beams of su?ciently high intensity.Thus we conclude that the LFV processesγe→τandγe→τare experimentally unobservable under the existing experimental limits onμ→eγandτ→eγ.In the other words the latter processes are much more sensitive to the LFV than the ones studied in the present paper.

IV.DISCUSSION AND CONCLUSIONS

We have studied LFV photoproduction ofμandτon atomic electrons.We extended the conventional formalism, for ordinary photoelectric e?ect with a?nal electron,to the case ofγe→μandγe→τprocesses.We have provided a general parametrization of the operator of electromagnetic current,instead of the commonly used parametrization of its matrix elements.This representation allowed us to consistently treat the o?-mass-shell initial atomic electron in terms of LFV analogs of the conventional monopole f M0,f E0and dipole f M1,f E1electromagnetic form factors of the electron.The studied LFV processes with real photons are independent of the monopole form factors,depending only

on the dipole LFV form factors f eμ

M1,E1,f eτM1,E1.These form factors are also involved inμ→eγandτ→eγdecays,

whose rates are limited by the existing experimental https://www.sodocs.net/doc/2c6662467.html,ing these experimental limits we extracted upper bounds on the dipole LFV form factors and predicted the total cross sections ofγe→μandγe→τprocesses.We also evaluated prospects for their experimental observation and arrived at a result that the event rate leaves no chance for this observation in any realistic experiment.In other words,the experiments looking forμ→eγandτ→eγdecays are much more sensitive to LFV than the above studied photoproduction processesγe→μandγe→τ.

Acknowledgments

This work was supported by CONICYT(Chile)under grant PBCT/No.285/2006.

[1]J.D.Vergados,Phys.Rep.133,1(1986);T.S.Kosmas,G.K.Leontaris,and J.D.Vergados,Prog.Part.Nucl.Phys.33,397

(1994);W.J.Marciano,”Lepton?avor violation,summary and perspectives”,Honolulu-Hawai,USA,October2-6,2000, https://www.sodocs.net/doc/2c6662467.html,/lepton

σ(e γ→μ)[f b ]

ω[GeV]10110010?110?2010?2210?2410?2610?28

10?30

10?32

10?34σ(e γ→τ)[f b ]ω[GeV]

10210110010?18

10?20

10?2210?2410?2610?2810?3010?32FIG.1:The total cross sections of the process γe →μ,τwith the lead Z =82atomic electron.The regions above the curves are excluded by the present experimental limits on μ,τ→eγdecays.